# Integral domain

2010 Mathematics Subject Classification: Primary: 13G05 [MSN][ZBL]

integral ring

A commutative ring with identity and without divisors of zero (cf. Zero divisor). Any field, and also any ring with identity contained in a field, is an integral domain. Conversely, an integral domain can be imbedded in a field. Such an imbedding is given by the construction of the field of fractions.

If \$A\$ is an integral domain, then the ring of polynomials \$A[X]\$ and the ring of formal power series \$A[[X]]\$ over \$A\$ are also integral domains. If \$A\$ is a commutative ring with identity and \$I\$ is any ideal in \$A\$, then the ring \$A/I\$ is an integral domain if and only if \$I\$ is a prime ideal. A ring \$A\$ without nilpotents is an integral domain if and only if the spectrum of \$A\$ is an irreducible topological space (cf. Spectrum of a ring).

Sometimes commutativity of \$A\$ is not required in the definition of an integral domain. Skew-fields and subrings of a skew-field containing the identity are examples of non-commutative integral domains. However, it is not true, in general, that an arbitrary non-commutative integral domain can be imbedded in a skew-field (see , and Imbedding of rings).

How to Cite This Entry:
Integral domain. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integral_domain&oldid=39096
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article